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IntroductionAnalytical Solutions
The Cloud ModelConclusions
Stochastic versus Uncertainty Modeling
Petra Friederichs, Michael Weniger, Sabrina Bentzien,
Andreas Hense
Meteorological Institute, University of Bonn
Dynamics and Statistic in Weather and Climate
Dresden, July 29-31 2009
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 1 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
MotivationOutline
Uncertainty Modeling
I Initial conditions – aleatoric uncertainty
Ensemble with perturbed initial conditions
I Model error – epistemic uncertainty
Perturbed physic and/or multi model ensembles
Simulations solving deterministic model equations!
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 2 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
MotivationOutline
Stochastic Modeling
I Stochastic parameterization – aleatoric uncertainty
Stochastic model ensemble
I Initial conditions – aleatoric uncertainty
Ensemble with perturbed initial conditions
I Model error – epistemic uncertainty
Perturbed physic and/or multi model ensembles
Simulations solving stochastic model equations!
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 3 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
MotivationOutline
Outline
Contrast both concepts
I Analytical solutions of simple damping equation
dv(t) = −µv(t)dt
I Simplified, 1-dimensional, time-dependent cloud model
I Problems
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 4 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Perturbed PhysicsStochastic ModelConclusions
Equation of Motion
Small particle of mass m with velocity v(t)
Subject to damping force F (t) = −µm · v(t)
dv(t) = −µv(t)dt
Solution of deterministic problem
v(t) = v0 exp(−µt)
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 5 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Perturbed PhysicsStochastic ModelConclusions
’Perturbed Physics’
γ is random variable
dv(t) = −γv(t)dt.
Solutions for different γ0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
t
v
deterministicnormaluniformgamma
γN ∼ N (µ, σ2) E [vN (t)] = v0 exp(− µt +
σ2
2t2)
γU ∼ U(µ±√
3σ)
E [vU (t)] = v0 exp(−µt)sinh(
√3σt)√
3σt
γΓ ∼ Γ(µ2
σ2,σ2
µ
)E [vΓ(t)] = v0
(1− −µt
µ2/σ2
)−µ2/σ2
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 6 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Perturbed PhysicsStochastic ModelConclusions
Stochastic Model
Damping from small-scale stochastic process
dv(t) = −µv(t)dt + v(t)σξdt,
ξt : Langevin force with
E [ξt ] = 0 and Cov[ξs , ξt ] = δ(t − s)
Stratonovich calculus
dvt = −µvtdt + σvt ◦ dWt
Solution
vt = v0 exp(− µt + σWt
)(exponential Brownian motion)
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 7 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Perturbed PhysicsStochastic ModelConclusions
Exponential Brownian motion
E[vt ] = v0 exp(−(µ− σ2
2
)t)
Var(vt
)= v2
0
[exp
(− 2(µ− σ2
)t)− exp
(− 2(µ− σ2
2
)t)]
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Three realisation of an exponential Brownian motion with µ=2, σ2=1
Time t
v t
det. solutionmean value
0 1 2 3 4 5 60
1
2
3
4
5
6Three realisation of an exponential Brownian motion with µ=2, σ2=3
Time t
v t
det. solutionmean value
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 8 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Perturbed PhysicsStochastic ModelConclusions
Non-delta Correlated Gaussian Noise
dv(t) = −µdt + v(t)εtv(t)dt
Solutionvt = v0 exp
(− µt +
∫ t
0εsds
)Noise εt with
E[εt ] = 0, Cov[εs , εt ] =D
2kexp(−k|t − s|).
is an Ornstein-Uhlenbeck process solving
dεt = −kεtdt +√
DdWt
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 9 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Perturbed PhysicsStochastic ModelConclusions
Non-delta Correlated Gaussian Noise
E[vt ] = v0 exp(− µt +
D
2k2
(t − 1− e−kt
k
))
D = 2k D = k2
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
t
v
deterministicdet. normalExpon. BrownianOU(k=0.01)OU(k=1)OU(k=100)
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
t
v
deterministicdet. normalExpon. BrownianOU(k=0.01)OU(k=1)OU(k=100)
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 10 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Perturbed PhysicsStochastic ModelConclusions
D = 2k D = k2
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 11 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Perturbed PhysicsStochastic ModelConclusions
’Perturbed physics’
E [v(t)]→∞ for t →∞ for all parameter distributions with
non-positive support
Stochastic model
Exponential Brownian motion:
I Non-stable solutions (σ2 > 2µ), increasing variance (σ2 > µ)
OU noise:
I Noise induced drift increases with 1/k (D = 2k)
I ’Perturbed physics’ solution for k → 0 (D = 2k)
I Exponential Brownian motion for k →∞ (D = k2)
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 12 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Model DescriptionPerturbed PhysicsStochastic Parameterization
1-dimensional Cloud Model
I Cylinder - constant radius
I Horizontally homogen in
constant environment
I Prognostic equations for
w(z), T (z), qv (z), qc(z),
qr (z)Fig: Dynamics of Multiscale Earth Systems,
C.Simmer(Eds.)
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 13 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Model DescriptionPerturbed PhysicsStochastic Parameterization
Koeln 04.07.1994 RR=11,723mm
Zeit in min
Hoe
he in
km
0.2
0.2
0.4
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2
0 10 20 30 40 50 60 70 80 90
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40mm/h
g/kg
0
1
2
3
4
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 14 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Model DescriptionPerturbed PhysicsStochastic Parameterization
0 = −1
ρ
∂(ρw)
∂z−
2
Ru
∂w
∂t=−w
∂w
∂z+
2
Rα2|we − w |(we − w) +
2
R(w − w)u +
„Tv − Tv,e
Tv,e
«g − (qc + qr )g
∂T
∂t=−w
∂T
∂z+
2
Rα2|we − w |(Te − T ) +
2
R(T − T )u − Γdw + SST
∂qv
∂t=−w
∂qv
∂z+
2
Rα2|we − w |(qv,e − qv ) +
2
R(qv − qv )u−
dqv
dt
∂qc
∂t=−w
∂qc
∂z+
2
Rα2|we − w |(qc,e − qc ) +
2
R(qc − qc )u −
dqc
dt
∂qr
∂t=−w
∂qr
∂z| {z }advection
+2
Rα2|we − w |(qr,eqr )| {z }turbulent entrainment
+2
R(qr − qr )u| {z }
dynamical entr.
−dqr
dt+ VR
∂r
∂z+
qr
ρ
∂(ρVR)
∂z| {z }sources
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 15 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Model DescriptionPerturbed PhysicsStochastic Parameterization
Micro-physical
processes after
Kessler (1969)...
Fig: von der Emde, Kahlig (1989)
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 16 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Model DescriptionPerturbed PhysicsStochastic Parameterization
Condensation:
qv → qc
− dqv
dt
VC
=qv − qvws
1 + εL2cqvws
cpRaT 2
· 1
∆tfor qv > qvws
Conversion:
qc → qr
− dqc
dt
CRAU
= kc(qc − qc,0) for qc > qc,0
qc,0 = 1ρ , kc = 10−3
Accretion:
qc → qr
− dqc
dt
CRAC
=π
4Eαrn0,rqc · λ−(3+βr ) · Γ(3 + βr )
E = 1 collection efficiency, αr , βr ;n0,r : Marshall-Palmer
λ−(3+βr ) =(
ρqr
πρpsn0,r
) (3+βr )4
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 17 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Model DescriptionPerturbed PhysicsStochastic Parameterization
Perturbed Physics
kc → kc (1 + η) E → E (1 + η)ra
in in
mm
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
−1 0 1 2
Conversion
rain
in m
m
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
−1 0 1 2
Accretion
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 18 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Model DescriptionPerturbed PhysicsStochastic Parameterization
Perturbed Physics
rain in mm
dens
ity
0.00
0.01
0.02
0.03
0.04
0.05
11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13 13.2 13.4
Conversion
rain in mm
dens
ity0.00
0.02
0.04
0.06
0.08
0.10
0.12
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Accretion
=
IntroductionAnalytical Solutions
The Cloud ModelConclusions
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 19 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Model DescriptionPerturbed PhysicsStochastic Parameterization
Stochastic Parameterization
(solved with Milstein Scheme)
kc → kc (1 + ξt) E → E (1 + ξt)
rain in mm
dens
ity
0.00
0.02
0.04
0.06
0.08
11.64 11.68 11.72 11.76 11.8
Conversion
rain in mm
dens
ity
0.00
0.01
0.02
0.03
0.04
0.05
11.2 11.4 11.6 11.8 12 12.2 12.4
Accretion
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 20 / 21
IntroductionAnalytical Solutions
The Cloud ModelConclusions
Conclusions and Outlook
I ’Perturbed physics’ ensembles do not represent realizations
from real process
I Stochastic parameterization may provide realistic distributions
I Solutions strongly depend on covariance function of noise (in
time and in space)
I Stochastic parameterizations should be derived from
microphysical processes
P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 21 / 21